A Generalized Kernel Estimate for Nonparametric Regression

Abstract

A nonparametric estimation procedure is proposed for estimating the regression function. The estimate can be viewed as a generalization of the classical kernel estimate. In practice, the classical estimate has some difficulties; one of them is the lack of having a simple or reliable procedure to select the bandwidth. The proposed estimate has several advantages due to its maintaining a simple linear model structure, which is similar to the one of the classical kernel estimate under the setting of circular design and equally spaced design points. The more reliable bandwidth selection procedure of Chiu (1990b) can be extended to select the bandwidth for the proposed estimate. Some other advantages, such as robust regression, weighted least squares estimation and adaptive smoothing, are also explored. Simulation studies were carried out to check and compare the performance of the proposed procedure with the classical kernel estimate and the local linear smoother with the bandwidth selected by various methods. The proposed procedure performs favorably well for all situations considered; it is often substantially better than the others when the regression function contains sharp peaks or when the design is not uniform. In addition to the simulation studies, the proposed procedures are applied to three real data sets. The applications demonstrates the ease of tailoring the proposed procedure to some specific situations.